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A257398 Expansion of phi(-x^6)^2 / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A257398 #13 Feb 16 2025 08:33:25
%S A257398 1,1,1,2,2,3,0,1,2,0,2,0,3,2,2,3,0,2,2,2,0,0,1,0,2,2,1,4,2,4,0,0,2,0,
%T A257398 4,1,0,0,4,2,1,0,2,2,0,0,0,2,2,4,2,1,2,4,2,2,0,1,0,0,4,0,2,4,0,0,0,2,
%U A257398 0,2,3,0,0,2,2,2,2,3,2,0,4,0,4,2,2,0,0
%N A257398 Expansion of phi(-x^6)^2 / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A257398 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257398 G. C. Greubel, <a href="/A257398/b257398.txt">Table of n, a(n) for n = 0..1000</a>
%H A257398 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257398 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257398 Expansion of phi(x^3) * f(x, x^2) in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
%F A257398 Expansion of q^(-1/24) * eta(q^2) * eta(q^6)^4 / (eta(q) * eta(q^12)^2) in powers of q.
%F A257398 Euler transform of period 12 sequence [1, 0, 1, 0, 1, -4, 1, 0, 1, 0, 1, -2, ...].
%e A257398 G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + x^7 + 2*x^8 + 2*x^10 + ...
%e A257398 G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + q^169 + 2*q^193 + ...
%t A257398 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] EllipticTheta[ 4, 0, x^6] ^2, {x, 0, n}];
%o A257398 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^4 / (eta(x + A) * eta(x^12 + A)^2), n))};
%K A257398 nonn
%O A257398 0,4
%A A257398 _Michael Somos_, Apr 21 2015